Distances to Lattice Points in Knapsack Polyhedra
Abstract
We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower bound.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- arXiv:
- arXiv:1805.04592
- Bibcode:
- 2018arXiv180504592A
- Keywords:
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- Mathematics - Combinatorics;
- 52C07;
- 90C10;
- 11H06;
- 11D04;
- 52B12